ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  smodm2 Unicode version

Theorem smodm2 5944
Description: The domain of a strictly monotone ordinal function is an ordinal. (Contributed by Mario Carneiro, 12-Mar-2013.)
Assertion
Ref Expression
smodm2  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )

Proof of Theorem smodm2
StepHypRef Expression
1 smodm 5940 . 2  |-  ( Smo 
F  ->  Ord  dom  F
)
2 fndm 5029 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
3 ordeq 4135 . . . 4  |-  ( dom 
F  =  A  -> 
( Ord  dom  F  <->  Ord  A ) )
42, 3syl 14 . . 3  |-  ( F  Fn  A  ->  ( Ord  dom  F  <->  Ord  A ) )
54biimpa 290 . 2  |-  ( ( F  Fn  A  /\  Ord  dom  F )  ->  Ord  A )
61, 5sylan2 280 1  |-  ( ( F  Fn  A  /\  Smo  F )  ->  Ord  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285   Ord word 4125   dom cdm 4371    Fn wfn 4927   Smo wsmo 5934
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-in 2980  df-ss 2987  df-uni 3610  df-tr 3884  df-iord 4129  df-fn 4935  df-smo 5935
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator